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  "body": "\n## Statement\n\nLet $n \\ge 1$ be an integer and let $1 \u003c p \u003c \\infty$.\nLet $\\Delta$ be the [[Laplacian|0E0B6RA8K0]] and let $\\norm{\\cdot}_{L^p}$ be the norm of the [[Lebesgue space|SASP14MA1Z]] $L^p(\\R^n)$.\nThere is $C = C(n, p)$ such that each $u \\in C_c^\\infty(\\R^n)$ satisfies\n$$\\norm{\\partial_i \\partial_j u}_{L^p(\\R^n)} \\le C \\norm{\\Delta u}_{L^p(\\R^n)} \\quad \\text{for all } i, j \\in \\set{1, \\ldots, n}.$$\n\n## Remarks\n\nOn the Fourier side, $\\widehat{\\partial_i \\partial_j u}(\\xi) = \\frac{\\xi_i \\xi_j}{\\abs{\\xi}^2} \\widehat{\\Delta u}(\\xi)$.\nSo $\\partial_i \\partial_j u = R_i R_j \\Delta u$, up to sign, where $R_i$ is the Riesz transform with multiplier $-i \\xi_i / \\abs{\\xi}$.\nEach $R_i R_j$ is a [[singular integral operator|HTMCEARGDD]], and Calderón–Zygmund theory shows that such operators are bounded on $L^p$ for $1 \u003c p \u003c \\infty$.\nThe proof of that bound uses the [[Calderón–Zygmund lemma|K1YKXVPVDG]] to get a weak type $(1,1)$ bound, then interpolation with the $L^2$ bound from Plancherel, then duality.\nThe estimate fails for $p = 1$ and for $p = \\infty$: a function with bounded Laplacian need not have bounded second derivatives.\nCombined with cut-off functions, the estimate gives the local $W^{2,p}$ estimates for solutions of $\\Delta u = f$, and by freezing coefficients for elliptic equations with continuous leading coefficients.\n\n## Explanations\n\n### L1 human\n\nIf you know how large the Laplacian of a function is, on average in the $L^p$ sense, then you also know how large all its second derivatives are.\nThe Laplacian is only one combination of second derivatives, so this is a surprising gain.\nIt holds for every exponent strictly between $1$ and $\\infty$, but not at the two ends.\n\n### L2 human\n\nThese estimates are the $L^p$ counterpart of the Schauder estimates in Hölder spaces.\nThey are the main tool to show that a weak solution of an elliptic equation with an $L^p$ right-hand side has second derivatives in $L^p$.\nThey are also a model case of the general theory of singular integrals, which treats operators like the Riesz and Hilbert transforms.\n",
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